We start by noting that \(\rho (\Gamma ^d)\)  was already computed in (REF ).
As \(\mathfrak {o}\)  is a Dedekind domain, we get that the intersection of infinitely many prime ideals in \(\mathfrak {o}\)  is the zero ideal. In particular, we have that the intersection of infinitely many ideals of the form \(\mathfrak {p} \setminus \mathfrak {p}^2\)  must be the empty set and hence, \(I=\emptyset \)  and \(O(H,\mathbb {E})_I = O(H, \mathbb {E})\) .
We need to check the assumptions of Theorem REF . Condition (REF ) follows directly from Lemma REF  applied to the polynomials \(f(x_0, \dots , x_d)=x_0\)  and \(g(x_0, \dots , x_d)=x_1\) .
Next we show that (REF ) is satisfied for any \(\alpha >0\) . Let \(A=(A_0, \dots , A_d)\in \mathfrak {o}^{d+1}\) , then we define \(f_A(x)= \sum _{j=0}^d A_{d-j} x^j\)  and we denote by \(\operatorname{disc}(f_A)\)  the discriminant of \(f_A\) . If \(\lbrace  \mathfrak {p} \in \mathcal {P}_K \ \mid \ A \in U_\mathfrak {p} \rbrace  = \emptyset \) , then (REF ) is trivially satisfied. Thus, we will now assume that \(\lbrace  \mathfrak {p} \in \mathcal {P}_K \ \mid \ A \in U_\mathfrak {p} \rbrace  \ne \emptyset \) .
As the discriminant is the Sylvester matrix of the resultant of \(f_A\)  and \(f_A^{\prime }\) , we get (as \(A_d\)  will always be multiplied by some \(A_j\)  with \(j\in \lbrace 0, \dots , d-1\rbrace \) )
\(\operatorname{disc}(f_A) \in \bigcap _{\mathfrak {p}\in \mathcal {P}_K, A \in U_\mathfrak {p}} \mathfrak {p}.\) 

Combining this with the observation that \(\operatorname{disc}(f_A)\ne 0\)  as \(f_A\)  is irreducible (as \(A\)  is contained in some \(U_\mathfrak {p}\) , \(f_A\)  satisfies the criterion of Eisenstein), we get
\( \begin{split}\vert N_{K/\mathbb {Q}}(\operatorname{disc}(f_A)) \vert &= N(\langle \operatorname{disc}(f_A)\rangle _{\mathfrak {o}}) \\&\ge \prod _{\mathfrak {p}\in \mathcal {P}_K, \ A \in U_\mathfrak {p}} \vert N(\mathfrak {p}) \vert \ge H^{\alpha \vert \lbrace  \mathfrak {p}\in \mathcal {P}_K \ \mid \ \mathfrak {p}\succeq H^\alpha , A \in U_\mathfrak {p} \rbrace  \vert },\end{split}\) 
where we denote by \(N\)  the absolute norm and \(N_{K/\mathbb {Q}}\)  the ideal norm on \(\mathfrak {o}\) .
On the other hand, using that the resultant is a homogeneous polynomial of degree \(2d-2\)  and \(N_{K/\mathbb {Q}}(\operatorname{disc}(f_A)) = \operatorname{det}_\mathbb {Z}(\mathfrak {o}\rightarrow \mathfrak {o}, x \mapsto \operatorname{disc}(f_A)x)\) , there exists a constant \(c>0\) , depending only on \(\mathbb {E}\) , such that for all \(H\ge 1\)  and all \(A\in \mathfrak {o}^{d+1}\)  holds
\( \vert N_{K/\mathbb {Q}}(\operatorname{disc}(f_A)) \vert \le c H^{(2d-2)k},\) 
where \(k=[K:\mathbb {Q}]\) .
Thus, combining (REF ) and (REF ) and taking the logarithm, we obtain for \(H\ge 1\)
\(\alpha \vert \lbrace  \mathfrak {p}\in \mathcal {P}_K \ \mid \ \mathfrak {p}\succeq H^\alpha , A \in U_\mathfrak {p} \rbrace  \vert \le \ln (c) + (2d-2) k.\) 

Hence, Condition (REF ) holds for any choice of \(\alpha >0\) .
Now we check Conditions (REF ) and (). Let \(\mathbb {E}\)  be an integral basis of \(\mathfrak {o}\) , \(k=[K:\mathbb {Q}]\)  and \(\varphi : \mathfrak {o}\rightarrow \mathbb {Z}^k\)  be the isomorphism of \(\mathbb {Z}\) -modules induced by the basis \(\mathbb {E}\) . We get
\(\frac{ \left|\bigcap _{j=1}^n U_{\mathfrak {p}_j} \cap O(H, \mathbb {E})^{d+1} \right|}{ (2H)^{(d+1)k}}\le \left( \frac{ \left|\bigcap _{j=1}^n \mathfrak {p}_j \cap O(H, \mathbb {E}) \right|}{(2H)^k} \right)^d.\) 

Hence, we are only interested in ideals of the form \( \bigcap _{j=1}^n \mathfrak {p}_j\) . In fact, as \(\varphi \)  preserves densities of lattices, it is enough to consider its image in \(\mathbb {Z}^k\)  under \(\varphi \) . By a similar argument as for [1]} one can show that for any \(n\in \mathbb {N}\)  there exists a universal constant \(c>0\)  such that for any lattice \(\Gamma \subseteq \mathbb {R}^n\)  of full rank and all \(H\in \mathbb {N}_{\ge 1}\)  holds
\(\left|\frac{\vert \Gamma \cap [-H, H)^n\vert }{(2H)^n} - \rho (\Gamma ) \right|\le c \rho (\Gamma ) \left( \frac{\operatorname{diam}(\Gamma )}{H} + \left( \frac{\operatorname{diam}(\Gamma )}{H} \right)^2 \right),\) 
where \(\operatorname{diam}(\Gamma )\)  denotes the diameter of the fundamental domain of \(\Gamma \) .
Hence, if \(\operatorname{diam}( \varphi ( \bigcap _{j=1}^n \mathfrak {p}_j)) \le cH\) , then we can estimate
\( \begin{split}&\frac{\left|\bigcap _{j=1}^n \mathfrak {p}_j \cap O(H, \mathbb {E}) \right|}{(2H)^k}= \frac{\left|\varphi \left(\bigcap _{j=1}^n \mathfrak {p}_j\right) \cap [-H,H)^k \right|}{(2H)^k}\le c^{\prime } \rho \left( \varphi \left(\bigcap _{j=1}^n \mathfrak {p}_j \right)\right) \\&= c^{\prime } \rho \left( \bigcap _{j=1}^n \mathfrak {p}_j \right)= c^{\prime } \prod _{j=1}^n [\mathfrak {o}: \mathfrak {p}_j]^{-1}= c^{\prime } \prod _{j=1}^n p_j^{-\operatorname{deg}(\mathfrak {p}_j)}\le c^{\prime } \prod _{j=1}^n \frac{1}{p_j}.\end{split}\) 
Thus, for \(d\ge 2\)  we can pick \(v_\mathfrak {p} = (1+c^{\prime })/p^d\)  and the series converges. All we need to show is that there exists some \(\alpha >0\)  and some universal constant \(C>0\)  such that for \(\mathfrak {p}_1, \dots , \mathfrak {p}_n \preceq H^\alpha \)  holds
\(\operatorname{diam}\left( \varphi \left(\bigcap _{j=1}^n \mathfrak {p}_j \right)\right) \le C H.\) 

However, by Minkowski's Second Theorem [2]} we have for any ideal \(\mathfrak {a} \in \mathfrak {o}\)
\(\operatorname{diam}(\varphi (\mathfrak {a}))\le \rho (\varphi (\mathfrak {a}))^{-1}= \rho (\mathfrak {a})^{-1}= [\mathfrak {o}: \mathfrak {a}].\) 

By the Chinese Remainder Theorem we get
\(\operatorname{diam}\left(\varphi \left( \bigcap _{j=1}^n \mathfrak {p}_j\right) \right)\le [\mathfrak {o}: \bigcap _{j=1}^n \mathfrak {p}_j]= \prod _{j=1}^n [\mathfrak {o}: \mathfrak {p}_j]\le \prod _{j=1}^n p_j^k,\) 

Hence, if \(\mathfrak {p}_1, \dots , \mathfrak {p}_n \preceq H^\alpha \) , then we get
\(\operatorname{diam}\left( \bigcap _{j=1}^n \mathfrak {p}_j \right)\le H^{\alpha n k}.\) 
Choosing \(\alpha = 1/(nk)\) , we obtain
\(\operatorname{diam}\left( \bigcap _{j=1}^n \mathfrak {p}_j \right)\le H.\) 
